identity element (thing)

The definition of a group requires that one element of each group, when combined with any other element of the group, yields that other element. If we symbolize the identity of a group G by e, we can state that for every element g in G,

Examples

In the addition group on the set of real numbers, the identity
element is 0, since for each real number r,

0 + r = r + 0 = r

Since addition for integers (or the rational numbers, or any number of subsets of the real numbers) forms a normal subgroup of addition for real numbers, 0 is the identity element for those groups, too.

In the multiplication group defined on the set of real numbers1, the identity
element is 1, since for each real number r,

1 * r = r * 1 = r

And of course, this also holds for the rational numbers and many other subgroups (multiplication for integers is not a group).

1For multiplication to form a group with the real numbers (or any subset), we have to remember to exclude the number 0.